Degenerate Chenciner Bifurcation Revisited

Abstract

Generic results for degenerate Chenciner (generalized Neimark–Sacker) bifurcation are obtained in the present work. The bifurcation arises from two-dimensional discrete-time systems with two independent parameters. We define in this work a new transformation of parameters, which enables the study of the bifurcation when degeneracy occurs. By the four bifurcation diagrams we obtained, new behaviors hidden by the degeneracy are brought to light.

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…In this study, the truncated normal form of the Chenciner bifurcation was analyzed in a degeneracy case, where the degeneracy condition is given by a 10 b 01 − a 01 b 10 = 0 and a 10 = a 01 = 0 or b 10 = b 01 = 0, as an answer to the problem open in [4,35].…”

“…As it is not possible to choose new coordinates β 1 , β 2 , the idea is to use only the initial parameters (α 1 , α 2 ). This leads to the modifications of the structure of the sets of points B 1,2 and C, thus obtaining concurrent lines at the origin, similar to the situation analyzed in other articles [10,15], but different from the cases studied in [4,35]. We want to specify how many bifurcation diagrams are obtained, many or few.…”

“…In that case, 32 bifurcation diagrams were obtained. A parallel approach to that of [4] is studied in [35] by using another regular transformation of parameters, where the product a 10 a 01 b 10 b 01 = 0. In the article [15], the functions β 1 (α) and β 2 (α) have a 10 = 0; a 01 = 0; b 10 = 0; b 01 = 0, obtaining four bifurcation diagrams.…”

“…[10] studied the case when a 20 = a 11 = a 02 = 0 and b 10 = 0, b 01 = 0 or a 20 = a 11 = a 02 = 0 and b 10 = 0, b 01 = 0, obtaining 18 different bifurcation diagrams. The stability of the fixed point O for |α| that is sufficiently small and, respectively, "the existence of closed invariant curves in the" [4] truncated normal form in all the cases was treated before [10,15,35].…”

“…The work is structured in six sections; after the Introduction (Section 1 and Appendices A and B), Section 2 presents the analysis of degenerate Chenciner bifurcation that "means the existence and stability of equilibrium points and invariant closed curves" [4] for this form of degeneracy, known as non-transversality, i.e., the "transformation of parameters is not regular at (0, 0)" [35]. In Section 3, it is described the existence of bifurcations curves and their dynamics in the parametric plane (α 1 , α 2 ) in Theorem 1.…”